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() =
{
()
}
u
Va
u
a
Œ
Va
and
is a regular po
int
of u
is called the image of V under u. The rational function u is said to be dominant if u( V )
is dense in W with respect to the Zariski topology.
It is easy to see that a rational function u : V Æ W is dominant if and only if W is
the smallest variety in W containing u( V ) (Exercise 10.13.6). The next proposition is
an immediate corollary of Proposition 10.13.10.
10.13.11. Proposition. The domain of definition of a rational function u : V Æ W
between affine varieties with V irreducible is an open set and the rational function
itself can be represented by an m-tuple
p
q
p
q
Ê
Ë
ˆ
¯
1
1
m
m
,...,
(10.84)
of functions, where p i , q i Πk[X] and the q i do not vanish on V .
Clearly a rational function on an irreducible variety is the special case of
where m is 1 and W = k above. This shows that it is natural to think of rational func-
tions on a variety as functions defined by quotients of polynomials. It follows
that polynomial functions are a special case of rational functions. They are of
course defined on all of V , but rational functions are strictly speaking not functions
in general because they may not be defined everywhere. However, Theorem
10.5.6 implies that in the case of an irreducible hypersurface they are defined every-
where except on possibly a finite set of points. Finally, in analogy with Proposition
10.13.8(2), it is easy to see that representatives for rational functions are not unique
in general.
It is worthwhile to see what all this means in the special case of hypersurfaces.
Let V be a hypersurface defined by an irreducible polynomial f. The denominators q j
in the representation (10.84) for the rational function u will then be polynomials
which are not divisible by f. Furthermore, if p/q and r/s are representatives for rational
functions u and v, respectively, which agree on V wherever they are both defined, then
ps - rq is divisible by f. The reason for this is that ps - rq vanishes on all but a finite
number of points of V and the result follows from Theorem 10.5.6.
Just as in the case of coordinate rings, let us see how function fields behave with
respect to maps. Unfortunately, rational functions are not necessarily defined every-
where and so, although the idea is the same as in the case of coordinate rings and
simple, the trick now is showing that everything is still well defined.
Let u : V Æ W be a rational function between irreducible affine varieties V and W .
Assume that u is represented by
p
q
p
q
Ê
Ë
ˆ
¯
1
1
m
m
,...,
,
where p i , q i Œ k[X] and the q i do not vanish on V . If f Œ k[ W ], then let f ° u denote the
rational function in k( V ) represented by
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